Multi-scale modeling and computation is an increasingly important area of
research with profound impact on computational science and applied
mathematics. Recently a number of techniques have been developed that
link the micro-scale and the macro-scale together in the same simulation.
We will discuss a computational framework that allows for the accuracy of
a micro-scale technique but with a computational cost that is closer to
that of a macro-scale method.

The nervous system produces rhythmic electrical activity in
many frequency ranges, and the rhythms displayed during waking
are tightly tied to cognitive state. This talk describes ongoing
work whose ultimate aim is to understand the uses of these
rhythms in sensory processing, cognition and motor control.
The method is to address the biophysical underpinnings of
the different rhythms and transitions among them, to get clues
to how specific important subsets of the cortex and hippocampus
process and transform spatio-temporal input. We focus in this
talk on the gamma rhythm (30-80 hz), which is associated with
attention and awareness, and theta (4-12), associated with
active exploration and learning of sequences. Via case studies,
we show that different biophysics corresponds to different
dynamical structure in the rhythms, with implications for
function. The mathematical tools come from dynamical systems,
and include the use of low-dimensional maps, probability and
geometric singular perturbations.

Composite
Materials: An Old Field of Study Full of New SurprisesSlides:html

Composite
materials have been studied for centuries, and have attracted
the interest of reknown scientists such as Poisson, Faraday,
Maxwell, Rayleigh, and Einstein. Their properties are usually
not just a linear average of the properties of the constituent
materials and can sometimes be strikingly different. The beautiful
red glass one sees in old church windows is a suspension of
small gold particles in glass. Sound waves travel slower in
bubbly water than in either water or air. In the last few
decades composites have been found to have some surprising
properties. Most materials, such as rubber, get thinner when
they are stretched, but it is possible to design composites
which get fatter as they are stretched. Electromagnetic signals
can travel faster in a composite than in the constituent phases.
It is possible to combine materials which expand when heated,
with voids, to obtain a material which contracts when heated.
It is still an open question as to what properties can be
achieved when one mixes two or more materials with known properties.
This lecture will survey some of the progress which has been
made and the role the IMA played in the development of the
field.

The
Riemann Hypothesis is now left as the most famous unsolved
problem in mathematics. Extensive computations of zeros have
been used not only to provide evidence for its truth, but
also for the truth of deeper conjectures that predict fine
scale statistics on the distribution of zeros of various zeta
functions. These conjectures connect number theory with physics,
and are regarded by many as the most promising avenue towards
a proof of the Riemann Hypothesis. However, as is often true
in mathematics, numerical data is subject to a variety of
interpretations, and it is possible to argue that the numerical
evidence we have gathered so far is misleading. Whatever the
truth may be, the computational exploration of zeros of zeta
functions is flourishing, and through projects such as the
ZetaGrid is drawing many amateurs into contact with higher
mathematics.

Array
imaging, like synthetic aperture radar and Kirchhoff migration
in seismic imaging, does not produce good reflectivity images
when there is clutter, or random scattering inhomogeneities,
between the reflectors and the array. Can the blurring effects
of clutter be controlled? I will discuss this issue in some
detail and explain why it is a central one for the recent
developments in the mathematics of imaging. I will also review
briefly the current status of array imaging and I will show
in particular that if array data is collected carefully, and
there is lots of it, then a good deal can be done to minimize
blurring by clutter.

Since their introduction in 1981,
viscosity solutions have become one of the fundamental tools
in the theory of nonlinear partial differential equations.
This lecture presents a brief history of the theory and the
main applications. Moreover it identifies two important future
directions (stochastic homogenization and stochastic partial
differential equations) which are related to the understanding
of the role of randomness in the theory of nonlinear equations
and related fields.

Over
the last 50 years we have seen a tremendous evolution in digital
signal processing. As computers become more and more powerful
they are able to deal with ever increasing amounts of digitized
media. So far we have witnessed three waves: audio (1D), images
(2D), and video (3D). Each wave of digitization comes with
its own need for algorithms and sets off a new branch of digital
signal processing. Today a forth wave in digital signal processing
is emerging: digital geometry processing. New technology exist
for quickly and accurately acquiring 3D geometry of objects:
A sub-millimeter digitization of Michelangelo's David for
example consists of over one billion samples. While audio,
images, and video are defined on Euclidean geometry and therefore
often used Fourier based algorithms, this no longer works
for digital geometry. We will describe new multiresolution
and wavelet based geometry representations and show how they
are used to build a digital geometry processing toolbox, including
denoising, filtering, editing, morphing, and compression.

During
the past twenty years, continuous optimization has become
(in the admittedly biased view of the speaker) ever more important
across applied mathematics, computer science, and real-world
applications of all kinds. This talk will survey selected
highlights, emphasizing two areas of active research (direct
search and interior-point methods) as well as the growing
association between optimization and other fields of applied
mathematics, computer science, science, engineering, and medicine
(such as partial differential equations, combinatorial optimization,
design, and data analysis).

Recent results illustrating state-of-the-art
analytic, geometric and probabilistic tecnhniques in the theory
of chaos will be presented, along with a discussion of the
limitations and potential applications of this theory as it
stands today.